abstract: Shape space is, e.g., Imm(S{n-1},Rn)Diff(S{n-1}). Geodesic equations of certain Riemann metrics on it are non-linear PDE's. Those coming from L2 metrics induce vanishing geodesic distance and are related to well-known ones like Burgers equation. Those coming from higher order Sobolev metrics are well posed and have somtimes explicit solutions.